Radio resource management in large wireless networks

ABSTRACT

A system for allocating resources in a communication network includes a plurality of access points and a central controller having a transceiver and a processor. Each of the plurality of access points is configured to identify traffic information and channel information and transmit the traffic information and the channel information to the central controller. The central controller is configured to receive the traffic information and the channel information from each of the plurality of access points and to determine resource allocation recommendations based at least in part on the received traffic information and the received channel information. The central controller is also configured to transmit the resource allocation recommendations to the plurality of access points. Each of the plurality of access points is configured to allocate a resource based on the resource allocation recommendations and on local network information.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims a priority benefit to U.S. ProvisionalPatent App. No. 62/461,441 filed on Feb. 21, 2017, the entire disclosureof which is incorporated herein by reference.

BACKGROUND

Recent years have witnessed an explosion of mobile data traffic due toproliferation of smart terminals (e.g., smart phones and tablet personalcomputers) and increasing use of mobile applications. It is predictedthat global mobile data traffic will increase eightfold between 2015 and2020. Wireless operators are investing heavily in infrastructure,including increasingly denser deployment of access points (APs) toimprove cellular coverage and capacity for homes and businesses inurban, suburban, as well as rural areas.

SUMMARY

An illustrative system for allocating resources in a communicationnetwork includes a plurality of access points and a central controllerhaving a transceiver and a processor. Each of the plurality of accesspoints is configured to identify its traffic information and channelinformation, and transmit the identified traffic information and thechannel information to the central controller. The central controller isconfigured to receive the traffic information and the channelinformation from each of the plurality of access points and to determineresource allocation recommendations based at least in part on thereceived traffic information and the received channel information. Theresource allocation recommendations are determined on a timescalemeasured in seconds, and each resource allocation recommendation isspecific to a given access point. The central controller is alsoconfigured to transmit the resource allocation recommendations to theplurality of access points. Each of the plurality of access points isconfigured to allocate a resource based on the resource allocationrecommendations and on local network information.

A method for allocating resources in a communication network includesreceiving, by a transceiver of a central controller, traffic informationand channel information from each of a plurality of access points. Themethod also includes determining, by a processor of the centralcontroller, resource allocation recommendations based at least in parton the received traffic information and the received channelinformation. Determining the resource allocation recommendationsincludes updating a candidate pattern set, where the candidate patternset includes all possible patterns, and where a pattern comprises asubset of the plurality of access points. Determining the resourceallocation recommendations also includes identifying one or morepatterns from the candidate pattern set that are to receive one or moreresources. Determining the resource allocation recommendations furtherincludes determining whether an optimality gap associated with theresource allocation recommendations is below a threshold. The methodfurther includes transmitting, by the transceiver, the resourceallocation recommendations to the plurality of access points.

Other principal features and advantages of the invention will becomeapparent to those skilled in the art upon review of the followingdrawings, the detailed description, and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Illustrative embodiments of the invention will hereafter be describedwith reference to the accompanying drawings, wherein like numeralsdenote like elements.

FIG. 1 depicts a high-level architecture of a system in accordance withan illustrative embodiment.

FIG. 2 is a flow diagram depicting operations performed by the system100 in accordance with an illustrative embodiment.

FIG. 3 is a high level flow diagram illustrating operations performed bythe pattern-pursuit algorithm in accordance with an illustrativeembodiment.

FIG. 4 depicts all patterns of a 3 AP, 2 UE network in accordance withan illustrative embodiment.

FIG. 5 depicts a toy network example with 3 APs and 2 UEs in accordancewith an illustrative embodiment.

FIG. 6 is a table depicting parameters compliant with the LTE standardin accordance with an illustrative embodiment.

FIG. 7 is a comparison of results of the present scheme with baselineschemes in accordance with an illustrative embodiment.

FIG. 8 is a diagram depicting the average packet delay versus packetarrival rate in accordance with an illustrative embodiment.

FIG. 9A depicts a deployment and user association for a large network inaccordance with an illustrative embodiment.

FIG. 9B is a topology graph corresponding to the marked area in FIG. 9Ain accordance with an illustrative embodiment.

FIG. 9C is an allocation graph corresponding to the marked area in FIG.9A in accordance with an illustrative embodiment.

FIG. 10 is a graph that compares the actual average packet delay of theproposed allocation scheme with the baseline schemes in accordance withan illustrative embodiment.

DETAILED DESCRIPTION

Described herein are a system and method that enable joint optimizationof radio resource allocation in an unprecedented scale. Simulationsdescribed below demonstrate the efficiency and effectiveness of the newtechnology in a network consisting of a thousand or more access pointsand several thousand user equipments (UEs).

With increasing number of smart terminals and widening use of mobileInternet applications, there has been an explosion of mobile traffic incommercial networks. To support the tremendous growth of data traffic, adense deployment of access points (APs) or small cells over a large areahas been considered as a promising candidate for future 5G networks. Theflexible multi-tier architecture can better match highly dynamic trafficdemands of UEs to possible serving APs. Due to irregularities of networktopology and sophisticated interference conditions, efficient spectrumallocation and user association becomes extremely crucial for harnessingthe full power of the infrastructure.

There have been many studies of resource management in cellularnetworks. In one study, a dynamic fractional frequency reuse scheme wasproposed to combat the inter-sector interference. In another study, aheuristic greedy search was proposed for user association. In otherstudies, a utility maximization framework and pricing-based associationmethods were proposed. The association problem has also been jointlyconsidered with resource allocation. However, in general, these studiesgive sub-optimal solutions either by solving a non-convex optimizationproblem for a small network or by running a distributed algorithm for alarge network, which is far from optimal.

To address resource management challenges, the inventors have derived anequivalent reformulation of the fundamental resource allocation and userassociation problem from the viewpoints of UEs. Such user-centricreformulation captures the fact that each UE's performance depends onlyon the interference pattern of no more than a constant number of APs inthe UE's neighborhood. This allows a low-complexity reformulation of theglobal problem, which reduces the total number of variables fromexponential to quadratic in the number of UEs. Specifically, describedherein is a pattern pursuit algorithm and proof that it can yield asolution within any given gap from the global optimum. The frameworkhere applies to all concave utility functions.

The goal of the present subject matter is to allocate resources in ametropolitan area network consisting of a very large number of APs. Thetotal overhead for a network controller to perform the proposed resourceallocation scheme is quite small since the timescale of resourceadaptation is considered to be once every few seconds or minutes. Forexample, the rate for sending 30,000 parameters (16 bits each) everyminute is only 8 kilobits per second (kbps). To validate the performanceof the proposed scheme, packet-level simulations are carried out. It hasbeen demonstrated that the proposed solution for networks with 1000 APsor more and 2500 UEs or more. It is observed that the proposed schemesignificantly outperforms other conventional schemes, and that theperformance is within a 7% gap from (an upper bound of) the globallyoptimal utility in a typical scenario.

Due to its simplicity, distributed AP-centric resource management hasbeen the dominant design principle in the first generation (1G) throughthe fourth generation (4G) cellular networks. Basically, each APoptimizes its performance metrics independently with little or no directinput from other APs. In most cases, APs/cells are carefully placed andconfigured so that mutual interference can be tolerated. However, withdenser deployment of APs, it becomes increasingly harder to place themat the most desirable locations due to leasing constraints and othergeographical constraints. Such dense deployment causes the trafficvariations in smaller cells to be much more pronounced, leading to morecomplex interference conditions. As a result, denser AP deployment alonedoes not lead to increased total network capacity. It is thereforecrucial for nearby APs to coordinate allocations of all radio resources,especially radio spectrum and power, so that mutual interference iseffectively managed.

Several coordinated resource management techniques have been developedin long term evolution (LTE) and LTE-Advanced (LTE-A) standards. Theseinclude inter-cell interference coordination (ICIC), enhanced ICIC(eICIC), and further enhanced ICIC (felCIC). In particular, eICIC letsan AP use almost blank subframes (ABS) to mitigate interference toneighboring cells. The coordination is accomplished throughcommunication between neighboring APs.

However, to date there has been no practical design that enables a largenumber of APs in a network covering a very large geographical area(hundreds of square kilometers or more) to fully coordinate theirphysical resource allocations. To address this, described herein aresystems and methods for large-scale joint optimization of radio resourcemanagement that can be scaled to a thousand APs/cells or more. Thetechnology is based on a dual-timescale optimization framework forspectrum allocation, user association, power control, and linkscheduling. On a relatively slow timescale (e.g., seconds or minutes), acentral controller collects traffic and channel state information fromall APs and periodically sends resource allocationinstructions/recommendations to the scheduler of each AP in the network.On a fast timescale (e.g., milliseconds), each AP exchanges localinformation (such as queue lengths and channel state information) withneighboring APs and schedules links and physical resource blocks (PRBs)based on the instructions/recommendations from the central controller aswell as information collected from the other APs.

This subject matter described herein is well matched to any resourcepooling designs such as the cloud radio access network (C-RAN). Bothresource allocation and link scheduling problems are formulated astractable optimization problems which can be solved efficiently at thecentral controller over slow timescale and at each AP over fasttimescale, respectively. Moreover, as discussed in more detail below,the solution obtained by the central controller is mathematicallyguaranteed to be within any given tolerance to the optimal allocation.The systems and methods of the present subject matter offer twoallocation schemes for selection, which are referred to as conservativeallocation and adaptive allocation. The adaptive allocation schemeperforms moderately better with additional computational complexity.

Typically, in a modern wireless communications system, such as a ThirdGeneration Partnership Project (3GPP) Long Term Evolution (LTE)compliant communications system, a number of cells or evolved NodeBs(eNB) (also commonly referred to as NodeBs, base stations, base terminalstations, communications controllers, network controllers, controllers,APs, and so on) may be arranged into a cluster of cells, with each cellhaving one or multiple transmit antennas. Additionally, each cell or eNBmay be serving a number of users (also commonly referred to as UserEquipments (UEs), wireless devices, mobile stations, users, subscribers,terminals, and so forth) based on a priority metric, such as fairness,proportional fairness, round robin, and the like, over a period of time.It is noted that the terms cell, AP, and eNB are often usedinterchangeably. For consistency herein, the network is assumed toconsist of a number of APs and a number of UEs.

FIG. 1 depicts a high-level architecture of a system 100 in accordancewith an illustrative embodiment. The system 100 in FIG. 1 includes threeAP clusters 105, 110, and 115, each of which includes a plurality ofAPs. It is to be understood that, in practice, each system will includea large number of AP clusters and that the use of only the three APclusters 105, 110, and 115 is for illustration purposes. Similarly, eachof the AP clusters may include a large number of individual APs. Asdepicted in FIG. 1, there is some overlap in the coverage areas of thevarious AP clusters such that some APs appear in multiple clusters. Theindividual APs in each of the AP clusters communicate with a centralcontroller 120 by way of a core network 125. The central controller 120can be any computing system, and includes at least a processor 121, amemory 122, and a transceiver 123 for exchanging information.

The individual APs in each of the AP clusters measure the traffic andchannel state information and interference conditions for itself and/orits neighbors, and transmit the information to the central controller120 periodically. The APs may also measure quality of service (QoS)information, a number of UEs being served, signal-to-noise ratios, linkusage, and any other network-related information for provision to thecentral controller 120. In each period, the central controller 120collects the information from all APs as the input, computes the mostdesirable resource allocations according to predetermined optimalcriteria, and distributes the corresponding allocationinstructions/recommendations to the APs. The predetermined optimalcriteria can be QoS, signal-to-noise ratio, and/or any other metricpertaining to network performance. In an illustrative embodiment, onlyinstructions/recommendations pertaining to a given AP are sent to thatAP. This periodic process is typically carried out on a relatively slowtimescale (e.g., seconds) in order to accommodate the latency ininformation gathering and computation time.

The information exchange between the individuals APs can be realizedthrough direct links between APs, which are often referred to asbackhauls. The information exchange between the individual APs and thenetwork controller 120 is the core network 125, which can also bereferred to as a backbone network, which is a wireline network. Theaggregate data rate of such exchanges is very small compared to thecapacity of the backhaul and backbone networks.

On a fast timescale, each AP exchanges queue length and channel stateinformation with neighboring APs and schedules links based on theexchanged information and on the recommended allocations from thecentral controller 120 in order to maximize the utilities of all itsneighboring APs. The utility of each AP can be general including the sumrate, minimum UE service rate (max-min fairness), and sum log-rate(proportional fairness). In an illustrative embodiment, the APs do notrely solely on the recommendations from the central controller 120because of the inherent latency involved in receiving therecommendations.

FIG. 2 is a flow diagram depicting operations performed by the system100 in accordance with an illustrative embodiment. In alternativeembodiments, fewer, additional, and/or different operations may beperformed. Additionally, the use of a flow diagram is not intended to belimiting with respect to the order of operations performed. Theoperations performed in block 200 are slow timescale operationsperformed at the central controller, and operations performed in block205 are fast timescale operations performed at an individual AP. In anoperation 210, the central controller receives traffic and channelinformation from all APs. In an operation 215, the central controllercomputes allocation schemes for all APs based on the receivedinformation. In an operation 220, the central controller sendsallocation recommendations to all APs.

In an operation 225, the AP exchanges queue length and channelinformation with its neighboring APs such that the APs can use thisinformation in determining whether to implement a recommendationreceived from the central controller. In an operation 230, the AP sendsits traffic and channel information to the central controller. In anoperation 235, the AP receives an allocation recommendation from thecentral controller. In an operation 240, the AP schedules links thatmaximize local network utility. In an illustrative embodiment, the APschedules the links based on both the recommendation from the centralcontroller and on the locally exchanged information from neighboringAPs.

The systems described herein are suitable for implementation in anyexisting centralized solution (typically in a small cluster), e.g., aC-RAN. Since the central controller collects information on a slowtimescale (once every a few seconds/minutes), which is much slower thanthe channel coherence time, the average channel conditions can beaccurately measured. In addition, the slow timescale also allows ampletime for channel state information exchange/feedback and jointoptimization of resource allocations in every period. In the meantime,since each AP only has to exchange queue length and channel stateinformation with neighboring APs, the overhead of exchanging informationon a fast timescale is insignificant in each individual cell.

The systems described herein can be implemented in a very large wirelessnetwork consisting of hundreds or thousands of APs, which may includemany APs and C-RANs as sub-networks. A single central controllercollects information from all or a large subset of the APs and computesefficient allocation instructions/recommendations for all or a subset ofthe APs. In a regional or nation-wide network, one or a few such centralcontrollers may be deployed in each metropolitan area.

On the slow timescale, the task of the central controller is todetermine which resource segments (e.g., frequency/time resources) toallocate to each AP, and subsequently, which sub-segments to allocate tospecific UEs associated with that AP. The goal is usually to maximizesome long-term network utility such as QoS. The technology is broadlyapplicable to arbitrary number of heterogeneous APs over an arbitrarygeographical area over arbitrary heterogeneous frequency bands (whichmay or may not be contiguous) using arbitrary heterogeneous physicallayer signaling. The technology applies to both the uplink and downlinktransmissions in a network. For concreteness and clarity, the presentdescription often uses a generic downlink transmission setting withseveral homogenous APs and homogeneous UEs using identical signalingover a single homogeneous frequency band.

The notion of a pattern is important to resource allocation in aninterference environment. A pattern refers to a subset of APs. In anetwork of n APs, there are 2^(n) distinct patterns, including an emptyone. A resource is considered to be allocated to a pattern if only APsin that pattern are allowed to use/share the resource. An allocation ofresources to all APs can be thought of as dividing the resources into2^(n) different pieces with various sizes, with each piece allocated toone pattern. In practice, it oftentimes occurs that all but a smallnumber of patterns are allocated zero resources.

Since each pattern corresponds to the set of transmitters in thedownlink, it determines the interference condition in the network, whichimplies the efficiency of each AP-UE link. The central controller shouldselect the best ones out of all 2^(n) patterns along with thecorresponding amounts of resources that maximize the network utility ofinterest. The spectrum allocation and user association problem at thecentral controller side can be formulated as a convex optimizationproblem with more than 2^(n) variables (formulation P1 below), which iscomputationally prohibitive in a large network with thousands of APs. Asopposed to such direct optimization formulation, introduced herein is acomputationally feasible problem formulation (see P2 below). Basically,the total number of variables and constraints in P2 increases onlylinearly with the number of UEs. The number of AP's that can serve a UEis assumed to be upper bounded by a constant, which is always the casein practice.

The tractable formulation P2 is obtained based in part on the fact thata UE can only be served by one or multiple APs from a subset of a smallnumber of APs near the UE (defined as candidate APs). This is due torapid signal attenuation over distance, which is referred to as pathloss and means that only a small number of APs are within range of thatUE. It suffices for each UE to only consider its candidate APs.Meanwhile, it is sufficient for each AP to only consider resourceallocation among APs that may interfere with its associated UEs. Thetractable formulation P2 uses only local variables defined inneighborhoods of UEs while maintaining the consistency of allocationsbetween overlapping neighborhoods.

The tractable formulation P2 can be applied to general large networksconsisting of different kinds of APs transmitting power at differentpower levels. The formulation exploits the hidden sparse structure ofthe UE-AP association. It also helps reduce the overhead of transmittingrecommended patterns to each AP since each AP only has to receive goodlocal patterns that exclude information about other APs that are faraway from it.

The tractable formulation P2 is a nonlinear mixed-integer optimizationproblem. Discussed in more detail below is a pattern-pursuit algorithm,described as Algorithm 1, for solving the optimization problem at thecentral controller. It is a first-order optimization algorithm. In eachiteration, the algorithm considers a linear approximation of theobjective function, and moves towards the maximizer of this linearfunction (taken over the same domain), which identifies a candidateallocation pattern. The spectrum allocation solution is chosen from allcandidate patterns. Since the central controller is solving a binarylinear programming in each iteration, due to the sparsity structure ofthe constraints, it can be solved quickly using standard algorithmsbased on branch and cut. Moreover, thanks to the convexity of theutility function, the optimality gap can be computed in each iterationas well. Hence, the algorithm can be terminated when the optimality gapis below a preset threshold. The resource allocation problem can besolved efficiently and near-optimally at the central controller on aslow timescale.

FIG. 3 is a high level flow diagram illustrating operations performed bythe pattern-pursuit algorithm in accordance with an illustrativeembodiment. In alternative embodiments, fewer, additional, and/ordifferent operations may be performed. Additionally, the use of a flowdiagram is not intended to be limiting with respect to the order ofoperations performed. In an operation 300, the pattern-pursuit algorithmis initialized by the central controller. In an operation 305, a linearapproximation of the objective function is optimized. In an operation310, a candidate pattern set is updated. Resources are allocated bychoosing patterns from the candidate pattern set in an operation 315. Inan operation 320, a determination is made regarding whether theoptimality gap is below a predetermined threshold. If it is determinedthat the optimality gap is not below the threshold, the process revertsagain to operation 305 and continues the process until the optimalitygap is below the threshold. If it is determined that the optimality gapis below the threshold, the pattern-pursuit algorithm is terminated inan operation 325.

The pattern-pursuit algorithm described herein fits the system structurebecause one recommended (or good) pattern is found at each iteration.Therefore, the central controller can keep sending recommended patternsto all APs after each iteration. Moreover, this also facilitates theupdating process within each AP.

Once the central controller obtains the near-optimal solution (i.e.,some recommended patterns and their corresponding weights), it sendsthese recommended patterns to each AP as a n instruction orrecommendation to its scheduler. Since each AP only interferes with itsneighboring APs, the instructions and/or recommendations to each AP mayonly include the allocation information for that AP and its neighboringAPs, with very small information exchange overhead.

The aforementioned allocation scheme is called conservative allocationin the sense that all the APs in a specific pattern are assumed to bealways interfering with each other. The advantage of the conservativeallocation is that there is no need for the scheduler (or whichever unitthat performs rate adaptation) to know the state of the other APsincluded in the pattern. However, the conservative rate is a lower boundon the actual rate because the interference is overestimated. Therefore,to seek a better analytic solution, an adaptive allocation scheme isalso introduced, which better models the interaction among APs.

The main idea of the adaptive allocation is the adoption of the notionof an active set instead of assuming all APs in a pattern are alwaysinterfering with each other. An active set is a set of APs that actuallytransmit signals. In a stable interactive queueing system, each activeset appears for a fraction of time (in other words, with certainprobability). Therefore, the queueing behavior of each UE is averagedover all possible active sets. An iterative algorithm alternatelyupdates the allocation variables and utilization variables are proposedto yield an adaptive allocation. For implementation, one may chooseeither the conservative allocation scheme or the adaptive allocationscheme, or a combination of the two in some form.

On a fast timescale, each AP schedules link transmissions following theinstructions/recommendations received from the central controller. Thelink/packet scheduling problem at the AP side is formulated as anoptimization problem. The inputs include the instantaneous queuelengths, the channel state information of all links within itsneighborhood, and the instructions/recommendations from the centralcontroller. At each time slot, each AP selects a local pattern tomaximize some local network utility. One embodiment is to use thepatterns from the central controller as the set of feasible patterns,and select one or a few local patterns with the best match to theinstructions/recommendations.

It is noted that that the formulated optimization problem of linkscheduling is similar to P1. The differences include: (1) The objectivefunction is the weighted sum rate with the weight being the queue lengthof each link; and (2) The optimization problem aims to choose oneoptimal pattern from the recommended patterns sent from the centralcontroller. The formulated optimization problem is a linear programmingwith finite feasible region, which can be solved by enumerating allrecommended patterns on a fast timescale.

Detailed Discussion of the Algorithm

The discussion below considers the downlink of a network consisting of nAPs and k UEs. A network controller is informed of the intensity ofindependent homogeneous Poisson traffic intended for every UE. Thecentral controller also receives sufficiently accurate reports ofchannel/interference information from all the APs. The resourceallocation is performed on a slow timescale, e.g., once every a fewseconds or minutes, which makes information exchange (channel stateinformation feedback) and joint resource allocation viable at thecentral controller. In addition, since the period of slow-timescaleresource allocation is much longer than the channel coherence time, theaverage channel conditions can be accurately modeled and measured usingpath loss and the statistics of small scale fading. The frequencyresources are assumed to be homogeneous on a slow timescale. Givenspectrum resource of bandwidth W Hertz (Hz), the task of the centralcontroller is to determine which spectrum segment(s) to allocate to eachAP-UE link in order to maximize the long-term network utility.

The set of AP indexes is denoted by N={1, . . . , n} and the set of UEindexes is denoted by K={1, . . . , k}. Arbitrary association is allowedsuch that each AP can simultaneously serve any subset of UEs and each UEcan be simultaneously served by any subset of APs. Furthermore, flexibleresource allocation is allowed in that each AP-UE link can use anarbitrary (possibly discontinuous) parts of the transmission spectrum.

A key aspect of total spectrum agility is the notion of the pattern,which as discussed above, refers to a subset of APs (i.e., a subset oftransmitters). A resource is said to be reserved for pattern A if theresource is to be shared by transmitters in A. As discussed herein,resources can refer to frequency resources. However, the present systemand method can be generalized to time, frequency, and/or other resourcessuch as spatial resources. In the downlink, a pattern A is a subset ofN, and APs in A are allowed simultaneous access to the frequency bandsreserved for pattern A. The pattern uniquely determines the interferencecondition and henceforth also the efficiency of every AP-UE link underthe pattern. There are 2^(n) distinct patterns in total, including theempty one. Because the spectrum is regarded as homogeneous on thetimescale of interest, the spectrum allocation problem can be formulatedas how to divide the spectrum among all 2^(n) patterns. As an exampleillustrated in FIG. 4, if there are three APs the spectrum can bedivided into 2³−1=7 segments. One segment is used by AP 1 exclusively(the pattern is {1}), a second is used by AP 2 exclusively (the patternis {2}), a third is used by AP 3 exclusively (the pattern is {3}), andthe remaining four segments include three shared by the pairs of APs(the patterns are {1,2}, {2,3}, and {1,3}, respectively), as well as onesegment shared by all three APs (the pattern is {1,2,3}).

The notion of the pattern is related to the concept of an independentset defined in the special case where the network is described by aweighted/unweighted conflict graph. In a conflict graph, since adjacentlinks cannot succeed simultaneously, it suffices to schedule onlypatterns corresponding to independent sets. The classical problem is tofind independent sets of links that maximize the network utility. It isassumed herein that nearby links cause soft interference rather thanhard conflict. The solution space therefore consists of all 2^(n)patterns, and as shown below the optimal solution consists of a verysmall subset of patterns.

The allocation problem can be divided into the following twosubproblems. The first allocation subproblem is to allocate bandwidthsto all 2^(n) patterns, denoted by a 2^(n)-dimensional vector:y=(y^(A))_(A⊂N), where y^(A)∈[0,1] is the fraction of bandwidth sharedby APs in A. It follows that:

$\begin{matrix}{{\sum\limits_{A \Subset N}^{\;}y^{A}} = 1.} & {{Eq}.\mspace{11mu} 1}\end{matrix}$

An efficient allocation allocates no resource to the empty pattern,yielding y^(Ø)=0. For every pattern A⊂N, every AP in A divides thespectrum reserved for A to serve all its associated UEs using orthogonalspectrum segments, which is the second allocation subproblem. Thebandwidth allocated to the link i→j (the link from AP i to UE j) overpattern A is dentoed as w_(i→j) ^(A). Consequently,

$\begin{matrix}{{{\sum\limits_{j \in K}w_{i\rightarrow j}^{A}} \leq y^{A}},{\forall{A \Subset N}},{i \in {A.}}} & {{Eq}.\mspace{11mu} 2}\end{matrix}$

As w_(i→j) ^(A) is only defined for i∈A, the result is exactly kn2^(n−1)such variables. Although the y variables specify the pattern bandwidthsonly, they directly imply a physical allocation as depicted in FIG. 4,which depicts all patterns of a 3 AP, 2 UE network in accordance with anillustrative embodiment. The allocations to the 2 UEs are revealed underpattern {1, 3}. Finer allocation to AP-UE links is then straightforward.As illustrated in FIG. 4, a physical spectrum allocation can be easilyassembled from the set of w variables satisfying Eq. 1 and Eq. 2. Also,UE j is associated to AP i if and only if w_(i→j) ^(A)>0 for somepattern A with i∈A.

For simplicity, it is assumed that each AP applies a flat power spectraldensity (PSD) over the allocated spectrum. The spectral efficiency oflink i→j over pattern A is denoted by s_(i→j) ^(A). It suffices todefine s_(i→j) ^(A) only for i∈A as s_(i→j) ^(A) is not used with i∉A inproblem formulations. To preempt any concern, the following relation isused:

s _(i→j) ^(A)=0,∀i∈N\A.  Eq. 3

Usually, the exclusive spectrum has higher spectral efficiency thanshared spectrum. In general, s_(i→j) ^(A)≥s_(i→j) ^(B) if ∈A⊂B. Thespectral efficiency s_(i→j) ^(A) can either be calculated based on pathloss and other impairments or be measured over time. For concreteness inobtaining numerical results, Shannon's formula is used for the linkefficiencies:

$\begin{matrix}{{s_{i\rightarrow j}^{A} = {\frac{W}{L}1\left( {i \in A} \right){\log_{2}\left( {1 + \frac{p_{i}g_{i\rightarrow j}}{n_{0} + {\sum\limits_{l \in {A:{l \neq i}}}^{\;}{p_{l}g_{l\rightarrow j}}}}} \right)}\mspace{14mu} {packets}\text{/}s}},} & {{Eq}.\mspace{11mu} 4}\end{matrix}$

where L is the average packet length in bits, p_(i) is the transmit PSDof AP i, n₀ is the noise PSD, g_(i→j) is the gain of link i→j whichcaptures the effects of path loss and shadowing, Σ_(l∈A:l≠i)p_(l)g_(l→j)is the interference received from other APs operating over the samepattern A, and the indicator function is defined as

$\begin{matrix}{{1(a)} = \left\{ \begin{matrix}1 & {{{if}\mspace{14mu} a\mspace{14mu} {holds}},} \\0 & {{if}\mspace{14mu} a\mspace{14mu} {does}\mspace{14mu} {not}\mspace{14mu} {{hold}.}}\end{matrix} \right.} & {{Eq}.\mspace{11mu} 5}\end{matrix}$

The link efficiency is normalized using factor W/L so that the units ofs_(i→j) ^(A) are packets/second. The service rate to UE j contributed byAP i∈A over pattern A is s_(i→j) ^(A)w_(i→j) ^(A). The total servicerate of UE j denoted as r_(j) can be calculated by summing over all APsover all patterns as follows:

$\begin{matrix}{r_{j} = {\sum\limits_{A \Subset N}^{\;}{\sum\limits_{i \in A}{s_{i\rightarrow j}^{A}{w_{i\rightarrow j}^{A}.}}}}} & {{Eq}.\mspace{11mu} 6}\end{matrix}$

The basic problem formulation is now described. The fundamental problemis to maximize the long-term utility by adapting the user associationand multi-pattern resource allocation. Collecting the constraints ofEqs. 1, 2, and 6, P0 is formulated as:

P 0a:  maximize_(r, w, y)u(r)${{P\; 0\; b\text{:}\mspace{11mu} {subject}\mspace{14mu} {to}\mspace{14mu} r_{j}} = {\sum\limits_{A \Subset N}^{\;}{\sum\limits_{i \in A}{s_{i\rightarrow j}^{A}w_{i\rightarrow j}^{A}}}}},{\forall{j \in K}}$${{P\; 0\; c\text{:}\mspace{11mu} {\sum\limits_{j \in K}w_{i\rightarrow j}^{A}}} \leq y^{A}},{\forall{A \Subset N}},{\forall{i \in A}}$${{P\; 0d\text{:}\mspace{11mu} \sum\limits_{A \Subset N}^{\;}} = 1},{{P\; 0e\text{:}\mspace{11mu} w_{i\rightarrow j}^{A}} \geq 0},{\forall{j \in K}},{\forall{A \Subset N}},{\forall{i \in A}}$

where u(r) is the network utility function, and y=(y^(A))_(A⊂N) andw=(w_(i→j) ^(A))_(j∈K,A⊂N,i⊂A) represent the bandwidth allocations. Thespectral efficiencies (s_(i→j) ^(A))_(j∈K,A⊂N,i∈A) are known parameters.Because the rate vector r=[r₁, . . . , r_(k)] is a linear transformationof the allocation vector w through P0b, the utility can be expresseddirectly as a function of the allocations: u(r(w)).

The P0 is a convex optimization problem as long as u(r) is concave in r.The sum rate, the minimum user service rate (max-min fairness), and thesum log-rate (proportional fairness) are all concave utility functions.The focus herein is on the average (negative) packet delay as thenetwork utility function:

$\begin{matrix}{{{u(r)} = {- {\sum\limits_{j \in K}\frac{\lambda_{j}}{\left( {r_{j} - \lambda_{j}} \right)^{+}}}}},} & {{Eq}.\mspace{11mu} 7}\end{matrix}$

where λ_(j) is the homogeneous Poisson packet arrival rate of UE j, andthe extended real-valued function 1/x⁺=1/x if >0 and 1/x⁺=+∞ if x≤0. Itcan be seen that 1/x⁺ is convex on (−∞, +∞). The choice of this utilityfunction also assumes exponential packet length and a conservative rate.If r_(j)≤h_(j), the packet delay is infinite, the system becomesunstable.

Theorem 1: There exists an optimal solution to P0 with at most k activepatterns, i.e., the optimal solution satisfies:

|{A⊂N|y ^(A)>0}|≤k.  Eq. 8

In addition, if the coefficients s_(i→j) ^(A) are drawn from a jointlycontinuous distribution, then, with probability 1, there are at most n−1UEs served by multiple APs in every optimal solution to P0. That is, theoptimal solution satisfies:

$\begin{matrix}\begin{matrix}{{{\left\{ {j \in K} \right.\mspace{14mu} {there}\mspace{14mu} {exist}\mspace{14mu} A_{1}},{A_{2} \Subset N},{i_{1} \in A_{1}},{i_{2} \in A_{2}}}} \\{{\left. {{{{that}\mspace{14mu} {satisfy}\mspace{14mu} i_{1}} \neq {i_{2}\mspace{14mu} {and}\mspace{14mu} w_{i_{1}\rightarrow j}^{A_{1}}}},{w_{i_{2}\rightarrow j}^{A_{2}} > 0}} \right\} } \leq {n - 1.}}\end{matrix} & {{Eq}.\mspace{11mu} 9}\end{matrix}$

The above Theorem 1, which has been proven, guarantees that although thetotal number of patterns grows exponentially with the number of APs inthe network, using a small number of patterns achieves the optimalperformance. Furthermore, it states that although we allow a UE to beserved by multiple APs, most UEs will be associated with only one AP inthe optimal solution.

Proposition 1: If the utility function u(r(w)) is affine in w, then themaximum utility in P0 can be attained by a single active pattern, whereeach AP serves only one UE. This proposition is proven below. A simpleexample for an affine utility function u(r(w)) is the weighted sum ratefunction. The above Proposition 1 admits a simple intuition: The utilityis contributed by a weighted sum of the bandwidths allocated to alllinks over all patterns, so shifting all resources to a dominant patterndoes not reduce the utility.

A scalable model and algorithm are now described. There arekn2^(n−1)+2^(n)+k variables in P0. P0 can be solved using a standardconvex optimization solver for networks with a small number of APs. Fora metropolitan area network consisting of hundreds or even thousands ofAPs, the space and time complexities of P0 become prohibitive. By firstdividing the network into many small clusters, one may solve forallocation in each cluster separately by assuming away the uncertaintiesabout inter-cluster interference. However, because interference fromoutside a cluster can penetrate deeply into a cluster, suchdivide-and-conquer solutions suffer significant loss. It is also notedthat any distributed solution is necessarily myopic and hence sufferssimilar loss. Thus, the network is treated in its entirety to develop ascalable, equivalent reformulation and an efficient near-optimal methodfor solving the new optimization problem is provided.

In a large network with many APs, a UE can in general only be served bya small subset of nearby APs due to path loss. For every j∈K, let N_(j)denote the set of APs whose received signal-to-noise ratios at UE j areabove a certain threshold ξ, i.e.,

$\begin{matrix}{\left. {N_{j} \equiv {\left\{ {i \in N} \right.\frac{p_{i}g_{i\rightarrow j}}{n_{0}}} > \xi} \right\}.} & {{Eq}.\mspace{11mu} 10}\end{matrix}$

N_(j) is referred to as the neighborhood of UE j. UE j treats all APsoutside N_(j) as stationary noise sources. This can be arbitrarilyprecise as N_(j) may be defined to include all APs except those receivedby UE j at well below the noise level. It is fair to assume the size ofall neighborhoods are upper bounded by a constant c₀, i.e., |N_(j)|≤c₀,∀j∈K. The choices of ξ of and c₀ in the actual numerical examples arediscussed below.

FIG. 5 depicts a toy network example with 3 APs and 2 UEs in accordancewith an illustrative embodiment. Here the neighborhood of UE 1 is {1,2}since AP 3 is far away from UE 1, making the received power from AP 3below the threshold ξ. Similarly, the neighborhood of UE 2 is {2,3}.Neighborhood N₁ can be thought of as a server of UE 1's traffic. Thepreceding assumptions imply that the efficiency of link i→j can benonzero only if i∈N_(j), and if so, it only depends on the activities ofAPs in the neighborhood of UE j. That is,

$\begin{matrix}{s_{i\rightarrow j}^{A} = {s_{i\rightarrow j}^{A\bigcap N_{j}}1\left( {i \in {A\bigcap N_{j}}} \right)}} & {{Eq}.\mspace{11mu} 11}\end{matrix}$

for all j∈K, A∈N, and i∈A. This is consistent with Eq. 3.

For every j∈K, all subsets of N_(j) constitute the set of local patternsof UE j. A new set of allocation variables (x_(i→j) ^(B)) are adopted,where for every j∈K, x_(i→j) ^(B) is only defined for B⊂N_(j) and i∈B.Here, x_(i→j) ^(B) denotes the bandwidth allocated to link i→j under thelocal pattern B, which can be obtained as:

$\begin{matrix}{{x_{i\rightarrow j}^{B} = {\sum\limits_{{A\bigcap{N:{A\bigcap N_{j}}}} = B}^{\;}{w_{i\rightarrow j}^{A}{\forall{j \in K}}}}},{B \Subset N_{j}},{i \in {B.}}} & {{Eq}.\mspace{11mu} 12}\end{matrix}$

That is, it is the sum bandwidth over all global patterns that match Bin the neighborhood of UE j. There are exactly

${\sum\limits_{j \in K}{{N_{j}}2^{{N_{j}} - 1}}} \leq {{kc}_{0}2^{c_{0} - 1}}$

such x variables. From the viewpoint of UE j, (x_(i→j) ^(B))_(B⊂N) _(j)_(,i∈B) describes how much bandwidth is allocated to all its associatedlinks over all its local patterns. Using Eqs. 11 and 12, the summationin P0b can be written as:

$\begin{matrix}{r_{j} = {\sum\limits_{A\bigcap N}^{\;}{\sum\limits_{i \in K}{s_{i\rightarrow j}^{A}w_{i\rightarrow j}^{A}}}}} & {{Eq}.\mspace{11mu} 13} \\{= {\sum\limits_{{A\bigcap{Ni}} \in}^{\;}{\sum\limits_{A\bigcap N_{j}}{s_{i\rightarrow j}^{A\bigcap N_{j}}w_{i\rightarrow j}^{A}}}}} & {{Eq}.\mspace{11mu} 14} \\{= {\sum\limits_{B\bigcap N_{j}}^{\;}{\sum\limits_{i \in B}{s_{i\rightarrow j}^{B}{\sum\limits_{{A\bigcap{N:{A\bigcap N_{j}}}} = B}^{\;}w_{i\rightarrow j}^{A}}}}}} & {{Eq}.\mspace{11mu} 15} \\{= {\sum\limits_{B\bigcap N_{j}}^{\;}{\sum\limits_{i \in B}{s_{i\rightarrow j}^{B}{x_{i\rightarrow j}^{B}.}}}}} & {{Eq}.\mspace{11mu} 16}\end{matrix}$

In Eq. 16, only local spectrum allocations x_(i→j) ^(B) with i∈B⊂N_(j)are used. Therefore, as substitutes of kn2n⁻¹ (global) w variables, atmost c₀2^(c) ⁰ ^(−l)=O(1) local x variables are involved in Eq. 16 for agiven j. This is sufficient as the sum over B⊂N_(j) and exhausts allpatterns of APs that may serve UE j.

As noted in the theorem above, there exists an optimal solution to P0that activates at most k patterns. Therefore, the local allocationvariables x should fit into k segments, where each segment representsthe allocation of one pattern. The set of all segment indexes is denotedby L={1, . . . , k}. By introducing replicas of the x variables in theform of (x_(i→j) ^(A,l))_(j∈K,l∈L,i∈A⊂N) _(j) , there is obtained anequivalent reformulation of P0, referred to as P1:

$\begin{matrix}{{maximize}_{r,x,d,h}{u(r)}} & {P\; 1a} \\{{{{subject}\mspace{14mu} {to}\mspace{14mu} r_{j}} = {\sum\limits_{A \Subset N_{j}}\; {\sum\limits_{i \in A}\; {s_{i\rightarrow j}^{A}{\sum\limits_{l \in L}\; x_{i\rightarrow j}^{A,l}}}}}},{\forall{j \in K}}} & {P\; 1b} \\{{x_{i\rightarrow j}^{A,l} \leq d_{j}^{A,l}},{\forall{j \in K}},{\forall{l \in L}},{\forall{A \Subset N_{j}}},{\forall{i \in A}}} & {P\; 1c} \\{{{d_{j}^{A,l} + {\sum\limits_{B \Subset {N_{m}:{{B\bigcap N_{j}} \neq {A\bigcap N_{m}}}}}d_{m}^{B,l}}} \leq 1},{\forall{l \in L}},{\forall{j \in K}},{\forall{A \Subset N_{j}}},{\forall{m \in {K:\; {{N_{m}\bigcap N_{j}} \neq \varnothing}}}},} & {P\; 1d} \\{{{\sum\limits_{j \in {K:{i \in N_{j}}}}\; {\sum\limits_{A \Subset {N_{j}:{i \in A}}}x_{i\rightarrow j}^{A,l}}} \leq h^{l}},{\forall{i \in N}},{\forall{l \in L}}} & {P\; 1d} \\{{{\sum\limits_{i \in L}\; h^{l}} \leq 1},} & {P\; 1f} \\{{d_{j}^{A,l} \in \left\{ {0,1} \right\}},{\forall{l \in L}},{\forall{j \in K}},{\forall{A \Subset N_{j}}}} & {P\; 1g} \\{{x_{i\rightarrow j}^{A,l} \geq 0},{\forall{l \in L}},{\forall{j \in K}},{\forall{A \Subset N_{j}}},{\forall{i \in {A.}}}} & {P\; 1h}\end{matrix}$

In P1, the spectrum is divided to k segments with bandwidths h¹, . . . ,h^(k), each corresponding to a global pattern. For the l-th segment, thevariables (x_(i→j) ^(A,l))_(j∈K,i∈A⊂N) _(j) and (d_(j) ^(A,l))_(j∈K,A⊂N)_(j) represent the allocation of this segment from all UEs' viewpoints.Here P1 b corresponds to Eq. 16. P1c implies that d_(j) ^(A,l) is theindicator of local pattern A from UE j's viewpoint over the l-thsegment, i.e., d_(j) ^(A,l)=1 if there exists i∈N_(j) such that x_(i→j)^(A,l)>0; otherwise d_(j) ^(A,l)=0. P1d constrains the consistency ofallocation over each segment among all UEs. That is, P1d enforces theallocation of no more than one pattern over segment l from every UE'sviewpoint. Compared with P0 which has O(kn2^(n)) variables, the numberof variables in P1 is

$\begin{matrix}{{{k{\sum\limits_{j \in K}\; {\left( {{N_{j}} + 2} \right)2^{{N_{j}} - 1}}}} + {2k}} = {{O\left( k^{2} \right)}.}} & {{Eq}.\mspace{14mu} 17}\end{matrix}$

Theorem 2: P0 and P1 are equivalent in the sense that they achieve thesame utility with identical rate vector(s). Moreover, given the optimalsolution to P1, the patterns and bandwidths of the optimal solution toP0 can be obtained as:

$\begin{matrix}{{A_{l} = {\bigcup\limits_{j \in K}{\bigcup\limits_{B \Subset {N_{j}:{d_{j}^{B,l} > 0}}}B}}},{\forall{l \in L}}} & {{Eq}.\mspace{14mu} 18} \\{{w_{i\rightarrow j}^{A_{l}} - x_{i\rightarrow j}^{A\bigcap N_{j}}},{\forall{l \in L}},{\forall{j \in K}},{\forall{i \in {A_{l}.}}}} & {{Eq}.\mspace{14mu} 19}\end{matrix}$

Theorem 2 is proven below. The following result is a useful buildingblock for an efficient algorithm for solving P1 to arbitrary precision.

Proposition 2: Suppose that u(r(x)) is an affine function of x. In termsof the maximum utility and the set of feasible (x, d), P1 is equivalentto P2:

$\begin{matrix}{{maximize}_{x,d}{u\left( {r(x)} \right)}} & {P2a} \\{{{{subject}\mspace{14mu} {to}\mspace{14mu} x_{i\rightarrow j}^{A}} \leq d_{j}^{A}},{\forall{j \in K}},{\forall{A \Subset N_{j}}},{\forall{i \in A}}} & {P\; 2\; b} \\{{{d_{j}^{A} + {\sum\limits_{B \Subset {N_{m}:{{B\bigcap N_{j}} \neq {A\bigcap N_{m}}}}}d_{m}^{B}}} \leq 1},{\forall{j \in K}},{\forall{A \in N_{j}}},{\forall{m \in {K:{{N_{m}\bigcap N_{j}} \neq \varnothing}}}},} & {P\; 2c} \\{{{\sum\limits_{j \in {K:{i \in N_{j}}}}\; {\sum\limits_{A \Subset N_{j}}x_{i\rightarrow j}^{A}}} \leq 1},{\forall{i \in N}}} & {P\; 2d} \\{{d_{j}^{A} \in \left\{ {0,1} \right\}},{\forall{j \in K}},{\forall{A \Subset N_{j}}}} & {P\; 2e} \\{{x_{i\rightarrow j}^{A} \in \left\{ {0,1} \right\}},{\forall{j \in K}},{\forall{A \Subset N_{j}}},{\forall{i \in {A.}}}} & {P\; 2f}\end{matrix}$

The key point of Proposition 2 is that when the utility function u(r(x))is affine in x, the optimal solution to P1 activates only one patternand each AP serves one UE that benefits the most, which yields asimplified formulation P2 (there is no need for the replica index l).More importantly, P2 is a binary lineary program (BLP) with only O(k)variables.

Although the mixed integer programming P1 has significantly fewervariables than the original problem P0 for a large network, it isnon-deterministic polynomial-time hard (NP-hard) in general. It is atleast as hard to compute the performance gap between an approximation ofP1 and the global optimal. The gap can, however, be upper bounded byoptimizing an upper bound of the utility function. A promising techniqueis then to iteratively optimize local linear expansions of the concaveutility function. In fact, because the expansion in each step must be anaffine upper bound, each step becomes a linear program.

For ease of notation, Λ is denoted as the feasible region in terms of xdefined by P2b-P2f and ν(x)=u(r(x)) denotes the concave utilityfunction. The ∇ν(x) is used to denote the gradient of ν(⋅).Specifically, if ν(⋅) is differentiable at x,

$\begin{matrix}{{\left\lbrack {\bigtriangledown \; {v(x)}} \right\rbrack_{i\rightarrow j}^{A} = \frac{\partial{v(x)}}{\partial x_{i\rightarrow j}^{A}}},{\forall{j \in K}},{A \Subset N_{j}},{i \in {A.}}} & {{Eq}.\mspace{14mu} 20}\end{matrix}$

If ν(⋅) is not differentiable at x, ∇ν(x) is minus the subgradient ofthe convex function −ν(⋅). For every q, x∈Λ, it is noted that

f _(q)(x)=ν(q)+

∇ν(q),x−q

  Eq. 21

where the inner product is defined in general as

$\begin{matrix}{{\langle{x,z}\rangle} = {\sum\limits_{j \in K}\; {\sum\limits_{A \Subset N_{j}}\; {\sum\limits_{i \in A}\; {x_{i\rightarrow j}^{A}{z_{i\rightarrow j}^{A}.}}}}}} & {{Eq}.\mspace{14mu} 22}\end{matrix}$

Due to its concavity, ν(⋅) must be upper bounded by its linearexpansion:

ν(x)≤f _(q)(x),∀x∈Λ.  Eq. 23:

Given a fixed feasible point q, one obtains an upper bound of the globalmaximum if u(r) is replaced in (P1a) by f_(q)(x). Since f_(q)(⋅) isaffine, the above proposition implies that a unique pattern can beidentified to maximize the affine approximation. Based on thisobservation, Algorithm 1 is proposed, which is an iterativepattern-pursuit algorithm for finding a solution within any given ε>0from the global optimum. Algorithm 1 is as follows:

Input: ε>0.

Output: x^((t)).

Initialization: t←0; P←Ø; pick an arbitrary pattern x⁽⁰⁾.

Repeat

Step 1. Compute x and d which maximize

∇ν(x^((t))), x

subject to constraints (P2b)-(P2f).

Step 2. t←t+1. If d∉P, P←P∪{d} otherwise, add an arbitrary new patternthat is not in P.

Step 3. Solve P1 by restricting to patterns in P and obtain the optimalallocation solution x^((t)).

until maximize_(x∈∇)

∇ν(x ^((t))),x−x ^((t))

<ε.

Algorithm 1 can be interpreted as a Frank-Wolfe type Algorithm (alsoknown as a conditional gradient algorithm). The main difference from aconventional algorithm is that instead of doing line search, Algorithm 1finds one recommended pattern in each iteration and re-optimizes P1using the set of recommended patterns identified so far. The recommendedpattern set P grows after each iteration because either one recommendedpattern is found or a random new pattern is added. In the worst case,Algorithm 1 takes no more than 2^(n) steps to terminate, because whenthe number of patterns in P reaches 2^(n), x^((t)) must be globallyoptimal, so that the condition to exit the loop must be met. As shown inthe numerical example below with 1000 APs and 2500 UEs, it takes onlyabout 50 steps to achieve an optimality gap of less than 7%.

Algorithm 1 has several important features. The algorithm hasrecommended pattern pursuit since Algorithm 1 starts with thefull-spectrum-reuse pattern in which all APs occupy the entire spectrum.In each iteration, it identifies one best pattern as the maximizer ofthis linear function (taken over the same domain). Due to the Theorem 1discussed above, it usually takes no more than k iterations to find theglobal optimum. The algorithm is also efficient in that the Step 1solves a BLP in the form of P2. Althought BLP is NP-complete in theworst case, many BLPs with sparse structure can be solved efficiently.As observed from numerical results, Step 1 takes a fairly small amountof time. In particular, if a branch and bound/cut method is used, theBLP step can be terminated as soon as a sufficiently tight upper boundis reached.

Additionally, Algorithm 1 has an optimality guarantee as stated inTheorem 3: Suppose Eq. 11 holds. For every ε>0, there exists a positiveinteger k such that ν(x^((k))) is at most ε away from the global optimumof P0.

Theorem 3 is proven in that P1 is always re-optimized using morepatterns than previous iterations, which results in non-decreasingseries of the utility (ν(x^((t))))_(t=0,1), . . . . This series mustconverge due to boundedness of the utility function. Let x* denote theglobal optimal. By Eqs. 21 and 23, it follows that:

ν(x*)−ν(x ^((t)))≤

∇ν(x ^((t))),x*−x ^((t))

^(.)  Eq. 24

Therefore, when the condition for terminating the loop in Algorithm 1 issatisfied, the optimality gap ν(x*)−ν(x^((k))) is guaranteed to be lessthan ε.

To obtain numerical results, parameters compliant with the LTE standardwere used. These parameters are depicted in the table set forth as FIG.6 in accordance with an illustrative embodiment. The maximum number ofpotential associations of a user is set as c₀=3. The results for theactual packet delay are obtained using a packet-level simulator, whichadapts the transmission time of each packet to the instantaneous activeAPs that are transmitting. It is noted that the delay of a packetincludes its transmission time and its waiting time in the queue. Theperformance gain of the proposed allocation schemes were investigated bycomparing them with the following baseline schemes: 1.Full-spectrum-reuse+maximum reference signal receive power (MaxRSRP):Every AP reuses all available spectrum and every UE is associated to thestrongest AP in terms of the received power. 2.Full-spectrum-reuse+optimal user association: Every AP reuses allavailable spectrum and user association is optimized for the utility. 3.A coloring algorithm. 4. Optimal lower bound: The optimal lower bound ofP0 obtained through Algorithm 1.

The performance of the proposed scheme was first considered inmedium-size networks, in which 100 APs and 200 UEs were randomly droppedover a 1100×1100 m² area. The average packet delay versus trafficintensity curves are shown in FIG. 7, which is a comparison with thebaseline schemes in accordance with an illustrative embodiment. As theaverage UE traffic increases to above 7.5 packets/second, all threebaseline schemes fail to support all the UEs. Conversely, the proposedsolution has significantly larger throughput (above 11 packets/second)than the other schemes. The proposed solution also significantly reducesthe delay especially in the high traffic regime. The reason is that theproposed solution adapts to the traffic conditions such that spectrum ismore reused more aggressively in the low traffic regime, and spectrumuse is more orthogonalized to avoid mutual interference. Furthermore,the curve of the lower bound of the optimum is quite close to the curveof the proposed scheme. This means the proposed solution is close to theglobal optimum of P0.

The present system and method were also tested in a large network.Specifically, the proposed scheme was used to compute near-optimalallocation for a network consisting of 1000 APs and 2500 UEs over a4200×4200 m² area. Since the coloring algorithm can not afford thecomputation in such large scale network, proposed scheme was comparedwith the first two baseline schemes.

The average packet delay versus packet arrival rate are shown in FIG. 8in accordance with an illustrative embodiment. In FIG. 8, each dottedcurve represents the average transmission time of the correspondingdelay curve with identical marker and color. The proposed solution hassignificantly larger throughput (above 21 packets/second) thanfull-spectrum reuse with maxRSRP association (7 packets/second) andfull-spectrum reuse with optimal user association allocation (14packets/second). The proposed solution also outperforms other schemes indelay especially in the high traffic regime. Furthermore, the proposedsolution is near optimal with less than 7% gap. Besides, compared withdelay, the transmission time increases much more slowly with trafficload, indicating that the spectrum is efficiently allocated to mitigateinterference among APs.

The obtained spectrum allocation and user association at average per UEpacket arrival rate of 20 packets/second is shown in FIGS. 9A-9C. FIG.9A depicts a deployment and user assoication for a large network inaccordance with an illustrative embodiment. FIG. 9B is a topogologygraph corresponding to the marked area in FIG. 9A in accordance with anillustrative embodiment. FIG. 9C is an allocation graph corresponding tothe marked area in FIG. 9A in accordance with an illustrativeembodiment. As shown in FIG. 9A, the lines connecting each UE-AP pairindicate an association. FIG. 9B shows the user association for themarked area. The numbers above each UE represent the UE index and itstraffic load, respectively. The number above each AP represents the APindex. The spectrum allocation for the marked area is shown in FIG. 9C.The widths of the rectangles represent fractions of the entire spectrumof the active patterns. The solid ones in each row are the spectrumsegments that are used by the corresponding AP to serve the UE whoseindex is marked on that spectrum segment. The algorithm achievestopology aware frequency reuse for interference management, as well asan efficient traffic aware spectrum allocation. Specifically, stronglyinterfering links (e.g., link 2→4 and link 3→5) are assigned differentspectrum segments, and the same spectrum segments are reused by twolinks that are far apart (e.g., link 10→25 and link 11→28). Moreover,UEs with light traffic loads or UEs on the transmission edge of two APs(e.g., UE 5) are assigned less spectrum, and vice versa.

To compare the theoretical delay with the actual delay, a packet-levelsimulator is used. The actual transmission rate of a resource reservedfor a pattern depends on the actual set of busy APs, which is a subsetof the pattern. FIG. 10 is a graph that compares the actual averagepacket delay of the proposed allocation scheme with the baseline schemesin accordance with an illustrative embodiment. In FIG. 10, each dottedcurve represents the average transmission time of the correspondingdelay curve with in accordance with the legend. Compared with thetheoretical results in FIG. 8, all three schemes achieve largerthroughput regions. That is because the service rate model of Eq. 6 isconservative, i.e., an AP's transmission rate over any spectrum segmentis the worst-case rate under the corresponding pattern, which is theachievable rate when all APs in the pattern are transmitting. This alsoexplains the fact that the proposed scheme is not as good as the secondbaseline scheme in the low traffic regime. But the proposed scheme stillachieves a quite larger throughput (31 packets/second/UE) than the otherschemes. Moreover, the delay is also significantly reduced by about 50%in the high traffic regime.

Thus, the systems and methods described herein effectively address thejoint user association and spectrum allocation problem in large networksover a slow timescale. A highly scalable reformulation of the networkutility maximization problem has been developed, and a pattern pursuitalgorithm is proposed which obtains near-optimal solution with anoptimality guarantee. As discussed above, the numerical results showsubstantial gains compared to all the other baseline schemes fornetworks with a large number of access points. The proposed algorithmapplies to any concave utility functions such as sum rate, minimum userservice rate (max-min fairness) and sum log-rate (proportionalfairness).

Proof of Proposition 1:

The affine utility function can be written as

$\begin{matrix}{{u\left( {r(w)} \right)} = {d + {\sum\limits_{j \in K}\; {\sum\limits_{A \Subset N}\; {\sum\limits_{i \in A}\; {c_{i\rightarrow j}^{A}w_{i\rightarrow j}^{A}}}}}}} & {{Eq}.\mspace{14mu} 25}\end{matrix}$

for some constants d and (c_(i→j) ^(A)). Then P0 can be rewritten as:

$\begin{matrix}{{{{maximize}\; y^{A}} \geq 0},{{\sum\limits_{A \Subset N}y^{A}} = {{1\mspace{14mu} {maximize}\; w_{i\rightarrow j}^{A}} \geq 0}},{{\sum\limits_{l \in K}\; w_{i\rightarrow l}^{A}} \leq {y^{A}{\forall{j \in K}}}},{\forall{A \Subset N}},{\forall{i \in {{A{\sum\limits_{j \in K}\; {\sum\limits_{A \Subset N}\; {\sum\limits_{i \in A}\; {c_{i\rightarrow j}^{A}w_{i\rightarrow j}^{A}}}}}} + {d.}}}}} & {{Eq}.\mspace{14mu} 26}\end{matrix}$

Define j*(i,A)∈K as a maximizer of c_(i→1) ^(A). It can be seen that thesolution to the inner problem in Eq. 26 is to let each AP serve thesingle UE with the largest weight for each pattern, i.e.,

w _(i→j) ^(A) =y ^(A)1(j=j*(i,A))  Eq. 27

for A⊂N, i∈A. Then Eq. 26 can be written as:

$\begin{matrix}{{{{maximize}\; y^{A}} \geq 0},{{\sum\limits_{A \Subset N}y^{A}} = {1{\sum\limits_{A \Subset N}\; {\sum\limits_{i \in A}\; {c_{i\rightarrow{j*{({i,A})}}}^{A}{y^{A}.}}}}}}} & {{Eq}.\mspace{14mu} 28}\end{matrix}$

Again, Eq. 28 can be solved by allocating all the resources to onepattern that has the largest weight, i.e., letting y^(A)*=1 where A*maximizes

$\sum\limits_{i \in A}{c_{i\rightarrow{j*{({i,A})}}}^{A}.}$

Proof of Theorem 1:

To prove Theorem 1, two additional equivalent optimization problems areintroduced as bridges between P0 and P1. P0 is equivalent to P4:

$\begin{matrix}{{maximize}_{r,w,y,h}{u(r)}} & {P\; 4a} \\{{{{subject}\mspace{14mu} {to}\mspace{14mu} r_{j}} = {\sum\limits_{A \Subset N}\; {\sum\limits_{i \in A}\; {s_{i\rightarrow j}^{A}{\sum\limits_{l \in L}\; x_{i\rightarrow j}^{A,l}}}}}},{\forall{j \in K}}} & {P\; 4b} \\{{{\sum\limits_{j \in K}w_{i\rightarrow j}^{A,l}} \leq y^{A,l}},{\forall{A \Subset N}},{\forall{i \in A}},{\forall{l \in L}}} & {P\; 4c} \\{{{\sum\limits_{A \Subset N}y^{A,l}} \leq h^{l}},{\forall{l \in L}}} & {P\; 4d} \\{{{\sum\limits_{A \Subset N}{y^{A,l}}_{0}} \leq 1},{\forall{l \in L}}} & {P\; 4e} \\{{{\sum\limits_{l \in L}h^{l}} \leq 1},} & {P4f} \\{{w_{i\rightarrow j}^{A,l} \geq 0},{\forall{l \in L}},{\forall{j \in K}},{\forall{i \in {A.}}}} & {P4g}\end{matrix}$

It is first shown that P0 is equivalent to P4 with constraint P4eremoved. To see this, it is recognized that the latter problem basicallysplits the variables in the former into k constituents in identicalform. The equivalence is then due to the concavity of the utilityfunction. To be precise, without P4e, if all variables with subscript lis set to 0 except for l=1, P4 reduces to P0. Thus P4 is a relaxation toP0. On the other hand, from any solution to P4 without constraint P4e,the variables of l=1, . . . , K can be combined to one feasible solutionof P0. Hence the equivalence.

It remains to show that the additional l₀ constraint P4e does not changethe optimal solution. As indicated above, P0 has an optimal solutionthat activates at most k patterns by Theorem 1. If the k active patternseach correspond to a distinct subscript l in P4, one obtains a feasiblesolution to P4 that yields the same utility. Specifically, if the kactive patterns found for P0 are A¹, . . . , A^(k)⊂N, and the optimal wand y variables are (w_(i→j) ^(A))_(j∈K,A⊂N,i∈A) and (y^(A))_(A⊂N). Thenthe variables of P4 are constructed as follows:

h ^(l) =y ^(A) ^(l)   Eq. 29

y ^(A,l) =y ^(A) ^(l) 1(A=A ^(l))  Eq. 30

w _(i→j) ^(A,l) =w _(i→j) ^(A) ^(l) 1(A=A ^(l))  Eq. 31

for l=1, . . . , k. Then it can be seen that all constraints in P4 aresatisfied and the same optimal utility is achieved. P4 is equivalent toP5:

$\begin{matrix}{{maximize}_{r,w,y,h}{u(r)}} & {P\; 5a} \\{{{{subject}\mspace{14mu} {to}\mspace{14mu} r_{j}} = {\sum\limits_{A \Subset N}\; {\sum\limits_{i \in A}\; {s_{i\rightarrow j}^{A}{\sum\limits_{l \in L}\; x_{i\rightarrow j}^{A,l}}}}}},{\forall{j \in K}}} & {P5b} \\{{w_{i\rightarrow j}^{A,l} \leq z_{j}^{A,l}},{\forall{A \Subset N}},{\forall{i \in A}},{\forall{l \in L}},{\forall{j \in K}}} & {P5c} \\{{{z_{j}^{A,l} + {\sum\limits_{B \Subset {N:{B \neq A}}}z_{m}^{B,l}}} \leq 1},{\forall{A \Subset N}},{\forall{l \in L}},{\forall j},{m \in K}} & {P5d} \\{{{\sum\limits_{j \in K}\; {\sum\limits_{A \Subset N}w_{i\rightarrow j}^{A,l}}} \leq h^{l}},{\forall{i \in N}},{\forall{l \in L}}} & {P5e} \\{{{\sum\limits_{l \in L}h^{l}} \leq 1},} & {P5f} \\{{z_{j}^{A,l} \in \left\{ {0,1} \right\}},{\forall{l \in L}},{\forall{j \in K}},{\forall{A \Subset N}}} & {P5g} \\{{w_{i\rightarrow j}^{A,l} \geq 0},{\forall{l \in L}},{\forall{j \in K}},{\forall{A \Subset N}},{\forall{i \in {A.}}}} & {P5h}\end{matrix}$

It is first noted that the utility functions of P4 and P5 are identical.Also, constraints P4b, P4f, and P4g are identical to constraints P5b,P5f, and P5h. Next, it is proven that every maximum of P4 is also amaximum of P5.

Suppose (r*, w*,y*, h*) is a maximum of P4. The variable z* is soughtsuch that (r*, w*, z*, h*) is feasible for P5. Fix l∈L. Constraint P4edictates that there is at most one active global pattern for every l∈L.Namely, one can identify one A_(l)*⊂N, such that y*^(B,l)=0 for everyB≠A_(l)*. From constraints P4c and P4d, it can be seen that:

$\begin{matrix}{{{\sum\limits_{j \in K}w_{i\rightarrow j}^{{*A_{l}^{*}},l}} \leq h^{*l}},} & {{Eq}.\mspace{14mu} 32} \\{{w_{i\rightarrow j}^{{*B},l} = 0},{\forall{B \neq {A_{l}^{*}.}}}} & {{Eq}.\mspace{14mu} 33}\end{matrix}$

For every i, j, l, and A, let z*_(j) ^(A,l)=0 if w_(i→j) ^(A,l)=0 andotherwise. Then by Eqs. 32 and 33, it can be seen that:

$\begin{matrix}{{z_{j}^{{*A_{l}^{*}},l} \leq 1},} & {{Eq}.\mspace{14mu} 34} \\{{z_{j}^{{*B},l} = 0},{\forall{B \neq {A_{l}^{*}.}}}} & {{Eq}.\mspace{14mu} 35}\end{matrix}$

It is apparent that these variables satisfy constraints P5c, P5d, andP5g. For the remaining P5e, it can be seen that:

$\begin{matrix}{{\sum\limits_{j \in K}{\sum\limits_{A \Subset N}w_{i->j}^{{*A},l}}} = {{\sum\limits_{j \in K}w_{i->j}^{{*A_{l}^{*}},l}} \leq {h^{*l}.}}} & {{Eq}.\; 36}\end{matrix}$

Therefore, (r*, w*, z*, h*) is feasible for P5. To show the converse, itis shown that if (r*, w*, z*, h*) is a maximum of P5, then there existsy* such that (r*, w*, y*, h*) is feasible for P4. Fix l∈L. ConstraintsP5d and P5g dictate that there is at most one active global pattern forall j∈K. Namely, one can identify one A_(l)*⊂N_(j), such that z*^(B,l)=0for every B≠A_(l)*. From constraints P5c, P5e, and P5h, it can be shownthat:

$\begin{matrix}{{w_{i->j}^{{*B},l} = 0},\mspace{25mu} {\forall{B \neq A_{l}^{*}}}} & {{Eq}.\; 37} \\{{\sum\limits_{j \in K}{\sum\limits_{A \Subset N}w_{i->j}^{{*A},l}}} = {{\sum\limits_{j \in K}w_{i->j}^{{*A_{l}^{*}},l}} \leq {h^{*l}.}}} & {{Eq}.\; 38}\end{matrix}$

If y* is defined as

${y^{{*A},l} = {\sum\limits_{j \in K}w_{i->j}^{{*A_{l}^{*}},l}}},$

then it follows that

$\begin{matrix}{{y^{{*B},l} = 0},\mspace{25mu} {\forall{B \neq A_{l}^{*}}}} & {{Eq}.\; 39} \\{y^{{*A_{l}^{*}},l} = {\sum\limits_{j \in K}{w_{i->j}^{{*A_{l}^{*}},l}.}}} & {{Eq}.\; 40}\end{matrix}$

By Eqs. 39 and 40, it is apparent that

$\begin{matrix}{{\sum\limits_{j \in K}w_{i->j}^{{*A},l}} \leq {y^{{*A},l}.}} & {{Eq}.\; 41}\end{matrix}$

In addition, it can be seen that

$\begin{matrix}{{{\sum\limits_{A \Subset N}y^{{*A},l}} = {y^{{*A_{l}^{*}},l} \leq h^{*l}}},{and}} & {{Eq}.\; 42} \\{{\sum\limits_{A \Subset N}{y^{{*A},l}}_{0}} = {{y^{{*A_{l}^{*}},l}}_{0} \leq 1.}} & {{Eq}.\; 43}\end{matrix}$

Therefore, these variables satisfy constraints P4c, P4d, and P4e. Hence(r*, w*, y*, h*) is also feasible for P4. It is concluded that everymaximum of P4 corresponds to a maximum of P5, and vice versa. Hence theequivalence of P4 and P5.

It can also be shown that P5 is equivalent to P1. The difference betweenP1 and P5 are entirely in the (w, z) variables associated with globalpatterns and (x, d) variables associated with local patterns. Theutilities P5a and P1a are identical. Constraints P5f and P1f areidentical. The global variables (w, z) are next related to the localvariables (x, d), so that feasibility of P1 and feasibility of P5 implyeach other.

It is shown that if (r, w, z, h) satisfy all constraints of P5, thenthere exist (x, d) such that (r, x, d, h) satisfy all constraints of P1.Let x and d variables be obtained as

$\begin{matrix}{{x_{i->j}^{A,l} = {\sum\limits_{{C \Subset {N:{C\bigcap N_{j}}}} = A}w_{i->j}^{C,l}}},{\forall{j \in K}},{l \in L},{A \in N_{j}},{i \in A}} & {{Eq}.\; 44} \\{{d_{j}^{A,l} = {\sum\limits_{{C \Subset {N:{C\bigcap N_{j}}}} = A}z_{j}^{C,l}}},{\forall{j \in K}},{l \in L},{A \in {N_{j}.}}} & {{Eq}.\; 45}\end{matrix}$

By P5c, P5g, and P5h, it is apparent that P1 c, P1 g, and P1h hold. Forevery UE_(j)∈K, every local pattern A∈N_(j), and every global patternC⊂N that satisfies C∩N_(j)=A, it can be seen that s_(i→j) ^(C)=s_(i→j)^(A) by Eq. 11. From P5b, Eqs. 11 and 44 indicate for every j∈K,

$\begin{matrix}{r_{j} = {\sum\limits_{C \Subset N}{\sum\limits_{i \in C}{s_{i->j}^{C}\; {\sum\limits_{l \in L}w_{i->j}^{C,l}}}}}} & {\mspace{265mu} {{Eq}.\; 46}} \\{= {\sum\limits_{A \Subset N_{j}}{\sum\limits_{{C \Subset {N:{C\bigcap N_{j}}}} = A}{\sum\limits_{i \in C}{s_{i->j}^{C}{\sum\limits_{l \in L}w_{i->j}^{C,l}}}}}}} & {{Eq}.\; 47} \\{= {\sum\limits_{A \Subset N_{j}}{\sum\limits_{i \in A}{s_{i->j}^{A}\; {\sum\limits_{l \in L}\left( {\sum\limits_{{C \Subset {N:{C\bigcap N_{j}}}} = A}w_{i->j}^{C,l}} \right)}}}}} & {{Eq}.\; 48} \\{= {\sum\limits_{A \Subset N_{j}}{\sum\limits_{i \in A}{s_{i->j}^{A}\; {\sum\limits_{l \in L}x_{i->j}^{A,l}}}}}} & {{Eq}.\; 49}\end{matrix}$

which is P1 b. Moreover, fix l∈L. Constraints P5d and P5g dictate thatthere is at most one active global pattern for all j∈L. Namely, one canidentify one A_(l)⊂N, such that z_(j) ^(B,l)=0 for every j∈K andB≠A_(l).

Next, the inequality P1d is examined where d is defined by Eq. 45. Forevery j, l, m, A, if A_(l)∩N_(j)=A, then

$\begin{matrix}{{d_{j}^{A,l} + {\sum\limits_{B \Subset {N_{m}:{{B\bigcap N_{j}} \neq {A\bigcap N_{m}}}}}d_{m}^{B,l}}} = {{\sum\limits_{{C \Subset {N:{C\bigcap N_{j}}}} = A}z_{j}^{C,l}} + {\sum\limits_{B \Subset {N_{m}:{{B\bigcap N_{j}} \neq {A\bigcap N_{m}}}}}{\sum\limits_{{D \Subset {N:{D\bigcap N_{m}}}} = B}{\hat{A}z_{m}^{D,l}}}}}} & {{Eq}.\; 50} \\{= {z_{j}^{A_{l},l} + {\sum\limits_{D \Subset {N:{{D\bigcap N_{m}\bigcap N_{j}} \neq {A_{l}\bigcap N_{m}\bigcap N_{j}}}}}z_{m}^{D,l}}}} & {{Eq}.\; 51} \\{\leq {z_{j}^{A_{l},l} + {\sum\limits_{D \Subset {N:{D \neq A_{l}}}}z_{m}^{D,l}}} \leq 1} & {{Eq}.\; 52}\end{matrix}$

where Eq. 52 is due to P5d. If A_(l)∩N_(j)≠A, then

$\begin{matrix}{{d_{j}^{A,l} + {\sum\limits_{B \Subset {N_{m}:{{B\bigcap N_{j}} \neq {A\bigcap N_{m}}}}}d_{m}^{B,l}}} = {{\sum\limits_{{C \Subset {N:{C\bigcap N_{j}}}} = A}z_{j}^{C,l}} + {\sum\limits_{B \Subset {N_{m}:{{B\bigcap N_{j}} \neq {A\bigcap N_{m}}}}}{\sum\limits_{{D \Subset {N:{D\bigcap N_{m}}}} = B}z_{m}^{D,l}}}}} & {{Eq}.\; 53} \\{\leq {0 + {\sum\limits_{D \Subset N}z_{m}^{D,l}}} \leq 1} & {{Eq}.\; 54}\end{matrix}$

where Eq. 54 is due to the special case of P5d with j=m. Therefore P1dis established. It remains to show Pie. By definition of Eq. 44,

$\begin{matrix}{{\sum\limits_{j \in K}{\sum\limits_{A \Subset N_{j}}x_{i->j}^{A,l}}} = {\sum\limits_{j \in K}{\sum\limits_{A \Subset N_{j}}{\sum\limits_{{C \Subset {N:{C\bigcap N_{j}}}} = A}w_{i->j}^{C,l}}}}} & {{Eq}.\; 55} \\{\leq {\sum\limits_{j \in K}{\sum\limits_{C \Subset N}w_{i->j}^{C,l}}} \leq h^{l}} & {{Eq}.\; 56}\end{matrix}$

where Eq. 56 is due to P5e. Thus (r, x, d, h) satisfy all constraintsP1b-P1h as long as (r, w, z, h) satisfy constraints P5b-P5h.

It is next shown that if (r, x, d, h) satisfy all constraints of P1,then there exists (w, z) such that (r, w, z, h) satisfy all constraintsof P5. The key is to reconstruct global variables (w, z) from localvariables (x, d). Fix l∈L. Constraints P1d and P1g dictate that there isat most one active local pattern in every neighborhood. Namely, forevery j∈K, we can identify one B_(j) ^(l)⊂N, such that d_(j) ^(B,l)=0for every B≠B_(j) ^(l). A global pattern can be defined as:

$\begin{matrix}{A_{l} = {\bigcup\limits_{j \in K}{B_{j}^{l}.}}} & {{Eq}.\; 57}\end{matrix}$

Due to P1d, it can be seen that A_(l)∩N_(j)=B_(j) ^(l). Define globalvariables:

$\begin{matrix}{w_{i->j}^{C,l} = {x_{i->j}^{B_{j}^{l},l}1\left( {C = A_{l}} \right)}} & {{Eq}.\; 58} \\{z_{j}^{C,l} = {d_{j}^{B_{j}^{l},l}1{\left( {C = A_{l}} \right).}}} & {{Eq}.\; 59}\end{matrix}$

Then P5g and P5h are trivial. Moreover,

$\begin{matrix}{r_{j} = {\sum\limits_{A \Subset N_{j}}{\sum\limits_{i \in A}{s_{i->j}^{A}{\sum\limits_{l \in L}x_{i->j}^{A,l}}}}}} & {{Eq}.\; 60} \\{= {\sum\limits_{i \in N_{j}}{\sum\limits_{l \in L}{s_{i->j}^{B_{j}^{l}}x_{i->j}^{B_{j}^{l},l}}}}} & {{Eq}.\; 61} \\{= {\sum\limits_{i \in N}{\sum\limits_{l \in L}{s_{i->j}^{A_{l}}w_{i->j}^{A_{l},l}}}}} & {{Eq}.\; 62} \\{{= {\sum\limits_{A \Subset N}{\sum\limits_{i \in A}{s_{i->j}^{A}{\sum\limits_{l \in L}w_{i->j}^{A,l}}}}}},} & {{Eq}.\; 63}\end{matrix}$

where Eq. 62 is due to Eq. 58. Therefore, P5b is established. P5c isestablished from P1c, Eq. 58, and Eq. 59. In addition, P5d isestablished due to P1d and Eq. 59. Finally, for P5e, it is shown that:

$\begin{matrix}{{\sum\limits_{j \in K}{\sum\limits_{A \Subset N}w_{i->j}^{A,l}}} = {\sum\limits_{j \in K}w_{i->j}^{A_{l},l}}} & {{Eq}.\; 64} \\{= {\sum\limits_{j \in K}x_{i->j}^{B_{j}^{l},l}}} & {{Eq}.\; 65} \\{\leq {\sum\limits_{j \in K}{\sum\limits_{A \Subset N_{j}}x_{i->j}^{A,l}}} \leq h^{l}} & {{Eq}.\; 66}\end{matrix}$

where Eq. 66 is due to P1 e. In all, the utility and constraints of P5are equivalent to those of P1. Hence the equivalence of the twooptimization problems.

The word “illustrative” is used herein to mean serving as an example,instance, or illustration. Any aspect or design described herein as“illustrative” is not necessarily to be construed as preferred oradvantageous over other aspects or designs. Further, for the purposes ofthis disclosure and unless otherwise specified, “a” or “an” means “oneor more”.

The foregoing description of illustrative embodiments of the inventionhas been presented for purposes of illustration and of description. Itis not intended to be exhaustive or to limit the invention to theprecise form disclosed, and modifications and variations are possible inlight of the above teachings or may be acquired from practice of theinvention. The embodiments were chosen and described in order to explainthe principles of the invention and as practical applications of theinvention to enable one skilled in the art to utilize the invention invarious embodiments and with various modifications as suited to theparticular use contemplated. It is intended that the scope of theinvention be defined by the claims appended hereto and theirequivalents.

What is claimed is:
 1. A system for allocating resources in acommunication network, the system comprising: a plurality of accesspoints, wherein each access point in the plurality of access points isconfigured to: identify traffic information and channel information; andtransmit the traffic information and the channel information to acentral controller; and the central controller, wherein the centralcontroller is configured to: receive, by a transceiver of the centralcontroller, the traffic information and the channel information fromeach of the plurality of access points; determine, by a processor of thecentral controller, resource allocation recommendations based at leastin part on the received traffic information and the received channelinformation, wherein the resource allocation recommendations aredetermined on a timescale measured in seconds, and wherein each resourceallocation recommendation is specific to a given access point; andtransmit, by the transceiver, the resource allocation recommendations tothe plurality of access points; and wherein each of the plurality ofaccess points is configured to allocate a resource based on the resourceallocation recommendations and on local network information.
 2. Thesystem of claim 1, wherein the resource allocation recommendationsspecify a spectrum segment to allocate to each access point-userequipment link.
 3. The system of claim 1, wherein the resourceallocation recommendations are configured to optimize quality ofservice.
 4. The system of claim 1, wherein the transceiver of thecentral controller is further configured to receive informationregarding an intensity of traffic intended for each user equipmentassociated with each of the plurality of access points.
 5. The system ofclaim 4, wherein the processor of the central controller is configuredto determine the resource allocation recommendations based in part onthe intensity of the traffic intended for each user equipment.
 6. Thesystem of claim 1, wherein the processor of the central controllerdetermines the resource allocation recommendations with a patternpursuit algorithm.
 7. The system of claim 6, wherein to implement thepattern pursuit algorithm, the processor of the central controller isconfigured to optimize a linear approximation of an objective function.8. The system of claim 7, wherein the objective function is a weightedsum rate, and wherein the weighted sum rate is weighed with a functionof a queue length of access point-user equipment links in thecommunications network.
 9. The system of claim 6, wherein the processorof the central controller is configured to update a candidate patternset, wherein the candidate pattern set includes all possible patterns,and wherein a pattern comprises a subset of the plurality of accesspoints.
 10. The system of claim 9, wherein the processor of the centralcontroller is configured to identify one or more patterns from thecandidate pattern set that are to receive one or more resources in orderto form the resource allocation recommendations.
 11. The system of claim10, wherein the processor of the central controller is furtherconfigured to determine whether an optimality gap associated with theresource allocation recommendations is below a threshold.
 12. The systemof claim 11, wherein the threshold for the optimality gap is apredetermined percentage.
 13. The system of claim 1, wherein theplurality of access points comprises at least one hundred access points,and wherein the at least one hundred access points are in communicationwith at least five hundred user equipments.
 14. A method for allocatingresources in a communication network, the method comprising: receiving,by a transceiver of a central controller, traffic information andchannel information from each of a plurality of access points;determining, by a processor of the central controller, resourceallocation recommendations based at least in part on the receivedtraffic information and the received channel information, and whereinthe determining includes: updating a candidate pattern set, wherein thecandidate pattern set includes all possible patterns, and wherein apattern comprises a subset of the plurality of access points;identifying one or more patterns from the candidate pattern set that areto receive one or more resources; and determining whether an optimalitygap associated with the resource allocation recommendations is below athreshold; and transmitting, by the transceiver, the resource allocationrecommendations to the plurality of access points.
 15. The method ofclaim 14, where the transmitting is performed responsive to adetermination that the optimality gap is below the threshold.
 16. Themethod of claim 14, wherein determining the resource allocationrecommendations comprises identifying a spectrum segment to allocate toeach access point-user equipment link.
 17. The method of claim 14,further comprising receiving, by the transceiver, information regardingan intensity of traffic intended for each user equipment associated witheach of the plurality of access points.
 18. The method of claim 17,wherein determining the resource allocation recommendations is performedbased in part on the intensity of the traffic intended for each userequipment.
 19. The method of claim 14, wherein the threshold is based ona desired level of quality of service.
 20. The method of claim 14,wherein the plurality of access points comprises at least one hundredaccess points, and wherein determining of the resource allocationrecommendations is performed on a timescale measured in seconds.